1. Installing the necessary packages
- GWmodel: for Geospatial Statistical modelling
- sf: for handling Spatial Data
- tidyverse: for Attribute data handling
- tmap: for plotting maps
packages = c('olsrr', 'corrplot', 'ggpubr', 'sf', 'spdep', 'GWmodel', 'tmap', 'tidyverse')
for (p in packages){
if(!require(p, character.only = T)){
install.packages(p)
}
library(p, character.only = T)
}
2. Geospatial Data Wrangling
Importing Geospatial Data
mpsz = st_read(dsn = "data/geospatial", layer = "MP14_SUBZONE_WEB_PL")
Reading layer `MP14_SUBZONE_WEB_PL' from data source
`C:\JunLonggggg\IS415\junlong-is415\_posts\2021-10-18-hands-on-exercise-9\data\geospatial'
using driver `ESRI Shapefile'
Simple feature collection with 323 features and 15 fields
Geometry type: MULTIPOLYGON
Dimension: XY
Bounding box: xmin: 2667.538 ymin: 15748.72 xmax: 56396.44 ymax: 50256.33
Projected CRS: SVY21Updating CRS information
mpsz_svy21 <- st_transform(mpsz, 3414)
st_crs(mpsz_svy21)
Coordinate Reference System:
User input: EPSG:3414
wkt:
PROJCRS["SVY21 / Singapore TM",
BASEGEOGCRS["SVY21",
DATUM["SVY21",
ELLIPSOID["WGS 84",6378137,298.257223563,
LENGTHUNIT["metre",1]]],
PRIMEM["Greenwich",0,
ANGLEUNIT["degree",0.0174532925199433]],
ID["EPSG",4757]],
CONVERSION["Singapore Transverse Mercator",
METHOD["Transverse Mercator",
ID["EPSG",9807]],
PARAMETER["Latitude of natural origin",1.36666666666667,
ANGLEUNIT["degree",0.0174532925199433],
ID["EPSG",8801]],
PARAMETER["Longitude of natural origin",103.833333333333,
ANGLEUNIT["degree",0.0174532925199433],
ID["EPSG",8802]],
PARAMETER["Scale factor at natural origin",1,
SCALEUNIT["unity",1],
ID["EPSG",8805]],
PARAMETER["False easting",28001.642,
LENGTHUNIT["metre",1],
ID["EPSG",8806]],
PARAMETER["False northing",38744.572,
LENGTHUNIT["metre",1],
ID["EPSG",8807]]],
CS[Cartesian,2],
AXIS["northing (N)",north,
ORDER[1],
LENGTHUNIT["metre",1]],
AXIS["easting (E)",east,
ORDER[2],
LENGTHUNIT["metre",1]],
USAGE[
SCOPE["Cadastre, engineering survey, topographic mapping."],
AREA["Singapore - onshore and offshore."],
BBOX[1.13,103.59,1.47,104.07]],
ID["EPSG",3414]]Extent of geospatial data, mpsz_svy21
st_bbox(mpsz_svy21)
xmin ymin xmax ymax
2667.538 15748.721 56396.440 50256.334 3. Aspatial Data Wrangling
Importing Aspatial data
Data are all in Km
condo_resale = read_csv("data/aspatial/Condo_resale_2015.csv")
Taking a look at what is in the aspatial data and confirming that it has been imported correctly:
glimpse(condo_resale)
Rows: 1,436
Columns: 23
$ LATITUDE <dbl> 1.287145, 1.328698, 1.313727, 1.308563,~
$ LONGITUDE <dbl> 103.7802, 103.8123, 103.7971, 103.8247,~
$ POSTCODE <dbl> 118635, 288420, 267833, 258380, 467169,~
$ SELLING_PRICE <dbl> 3000000, 3880000, 3325000, 4250000, 140~
$ AREA_SQM <dbl> 309, 290, 248, 127, 145, 139, 218, 141,~
$ AGE <dbl> 30, 32, 33, 7, 28, 22, 24, 24, 27, 31, ~
$ PROX_CBD <dbl> 7.941259, 6.609797, 6.898000, 4.038861,~
$ PROX_CHILDCARE <dbl> 0.16597932, 0.28027246, 0.42922669, 0.3~
$ PROX_ELDERLYCARE <dbl> 2.5198118, 1.9333338, 0.5021395, 1.9910~
$ PROX_URA_GROWTH_AREA <dbl> 6.618741, 7.505109, 6.463887, 4.906512,~
$ PROX_HAWKER_MARKET <dbl> 1.76542207, 0.54507614, 0.37789301, 1.6~
$ PROX_KINDERGARTEN <dbl> 0.05835552, 0.61592412, 0.14120309, 0.3~
$ PROX_MRT <dbl> 0.5607188, 0.6584461, 0.3053433, 0.6910~
$ PROX_PARK <dbl> 1.1710446, 0.1992269, 0.2779886, 0.9832~
$ PROX_PRIMARY_SCH <dbl> 1.6340256, 0.9747834, 1.4715016, 1.4546~
$ PROX_TOP_PRIMARY_SCH <dbl> 3.3273195, 0.9747834, 1.4715016, 2.3006~
$ PROX_SHOPPING_MALL <dbl> 2.2102717, 2.9374279, 1.2256850, 0.3525~
$ PROX_SUPERMARKET <dbl> 0.9103958, 0.5900617, 0.4135583, 0.4162~
$ PROX_BUS_STOP <dbl> 0.10336166, 0.28673408, 0.28504777, 0.2~
$ NO_Of_UNITS <dbl> 18, 20, 27, 30, 30, 31, 32, 32, 32, 32,~
$ FAMILY_FRIENDLY <dbl> 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, ~
$ FREEHOLD <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, ~
$ LEASEHOLD_99YR <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ~head(condo_resale$LONGITUDE)
[1] 103.7802 103.8123 103.7971 103.8247 103.9505 103.9386head(condo_resale$LATITUDE)
[1] 1.287145 1.328698 1.313727 1.308563 1.321437 1.314198summary(condo_resale)
LATITUDE LONGITUDE POSTCODE SELLING_PRICE
Min. :1.240 Min. :103.7 Min. : 18965 Min. : 540000
1st Qu.:1.309 1st Qu.:103.8 1st Qu.:259849 1st Qu.: 1100000
Median :1.328 Median :103.8 Median :469298 Median : 1383222
Mean :1.334 Mean :103.8 Mean :440439 Mean : 1751211
3rd Qu.:1.357 3rd Qu.:103.9 3rd Qu.:589486 3rd Qu.: 1950000
Max. :1.454 Max. :104.0 Max. :828833 Max. :18000000
AREA_SQM AGE PROX_CBD PROX_CHILDCARE
Min. : 34.0 Min. : 0.00 Min. : 0.3869 Min. :0.004927
1st Qu.:103.0 1st Qu.: 5.00 1st Qu.: 5.5574 1st Qu.:0.174481
Median :121.0 Median :11.00 Median : 9.3567 Median :0.258135
Mean :136.5 Mean :12.14 Mean : 9.3254 Mean :0.326313
3rd Qu.:156.0 3rd Qu.:18.00 3rd Qu.:12.6661 3rd Qu.:0.368293
Max. :619.0 Max. :37.00 Max. :19.1804 Max. :3.465726
PROX_ELDERLYCARE PROX_URA_GROWTH_AREA PROX_HAWKER_MARKET
Min. :0.05451 Min. :0.2145 Min. :0.05182
1st Qu.:0.61254 1st Qu.:3.1643 1st Qu.:0.55245
Median :0.94179 Median :4.6186 Median :0.90842
Mean :1.05351 Mean :4.5981 Mean :1.27987
3rd Qu.:1.35122 3rd Qu.:5.7550 3rd Qu.:1.68578
Max. :3.94916 Max. :9.1554 Max. :5.37435
PROX_KINDERGARTEN PROX_MRT PROX_PARK
Min. :0.004927 Min. :0.05278 Min. :0.02906
1st Qu.:0.276345 1st Qu.:0.34646 1st Qu.:0.26211
Median :0.413385 Median :0.57430 Median :0.39926
Mean :0.458903 Mean :0.67316 Mean :0.49802
3rd Qu.:0.578474 3rd Qu.:0.84844 3rd Qu.:0.65592
Max. :2.229045 Max. :3.48037 Max. :2.16105
PROX_PRIMARY_SCH PROX_TOP_PRIMARY_SCH PROX_SHOPPING_MALL
Min. :0.07711 Min. :0.07711 Min. :0.0000
1st Qu.:0.44024 1st Qu.:1.34451 1st Qu.:0.5258
Median :0.63505 Median :1.88213 Median :0.9357
Mean :0.75471 Mean :2.27347 Mean :1.0455
3rd Qu.:0.95104 3rd Qu.:2.90954 3rd Qu.:1.3994
Max. :3.92899 Max. :6.74819 Max. :3.4774
PROX_SUPERMARKET PROX_BUS_STOP NO_Of_UNITS
Min. :0.0000 Min. :0.001595 Min. : 18.0
1st Qu.:0.3695 1st Qu.:0.098356 1st Qu.: 188.8
Median :0.5687 Median :0.151710 Median : 360.0
Mean :0.6141 Mean :0.193974 Mean : 409.2
3rd Qu.:0.7862 3rd Qu.:0.220466 3rd Qu.: 590.0
Max. :2.2441 Max. :2.476639 Max. :1703.0
FAMILY_FRIENDLY FREEHOLD LEASEHOLD_99YR
Min. :0.0000 Min. :0.0000 Min. :0.0000
1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.:0.0000
Median :0.0000 Median :0.0000 Median :0.0000
Mean :0.4868 Mean :0.4227 Mean :0.4882
3rd Qu.:1.0000 3rd Qu.:1.0000 3rd Qu.:1.0000
Max. :1.0000 Max. :1.0000 Max. :1.0000 Seems that the scale of some variables are not comparable, this may require some data processing before using GWR.
4. Converting aspatial data frame into sf object
condo_resale.sf <- st_as_sf(condo_resale,
coords = c("LONGITUDE", "LATITUDE"),
crs=4326) %>%
st_transform(crs=3414)
head(condo_resale.sf)
Simple feature collection with 6 features and 21 fields
Geometry type: POINT
Dimension: XY
Bounding box: xmin: 22085.12 ymin: 29951.54 xmax: 41042.56 ymax: 34546.2
Projected CRS: SVY21 / Singapore TM
# A tibble: 6 x 22
POSTCODE SELLING_PRICE AREA_SQM AGE PROX_CBD PROX_CHILDCARE
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 118635 3000000 309 30 7.94 0.166
2 288420 3880000 290 32 6.61 0.280
3 267833 3325000 248 33 6.90 0.429
4 258380 4250000 127 7 4.04 0.395
5 467169 1400000 145 28 11.8 0.119
6 466472 1320000 139 22 10.3 0.125
# ... with 16 more variables: PROX_ELDERLYCARE <dbl>,
# PROX_URA_GROWTH_AREA <dbl>, PROX_HAWKER_MARKET <dbl>,
# PROX_KINDERGARTEN <dbl>, PROX_MRT <dbl>, PROX_PARK <dbl>,
# PROX_PRIMARY_SCH <dbl>, PROX_TOP_PRIMARY_SCH <dbl>,
# PROX_SHOPPING_MALL <dbl>, PROX_SUPERMARKET <dbl>,
# PROX_BUS_STOP <dbl>, NO_Of_UNITS <dbl>, FAMILY_FRIENDLY <dbl>,
# FREEHOLD <dbl>, LEASEHOLD_99YR <dbl>, geometry <POINT [m]>5. Exploratory Data Analysis
EDA using Stats graphs
ggplot(data=condo_resale.sf, aes(x=`SELLING_PRICE`)) +
geom_histogram(bins=20, color="black", fill="light blue")
The selling price distribution can be seen to be right skewed, which could mean that more condominium units were sold at relative lower prices or that there are outliers. However in this case, we just need to normalize the scale using a log transformation.
condo_resale.sf <- condo_resale.sf %>%
mutate(`LOG_SELLING_PRICE` = log(SELLING_PRICE))
Let’s take a look at the distribution again
ggplot(data=condo_resale.sf, aes(x=`LOG_SELLING_PRICE`)) +
geom_histogram(bins=20, color="black", fill="light blue")
Now the distribution is less skewed and even starts to resembles a gaussian distribution.
Multiple Histogram Plots distribution of variables (aka Pair plots)
AREA_SQM <- ggplot(data=condo_resale.sf, aes(x= `AREA_SQM`)) +
geom_histogram(bins=20, color="black", fill="light blue")
AGE <- ggplot(data=condo_resale.sf, aes(x= `AGE`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_CBD <- ggplot(data=condo_resale.sf, aes(x= `PROX_CBD`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_CHILDCARE <- ggplot(data=condo_resale.sf, aes(x= `PROX_CHILDCARE`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_ELDERLYCARE <- ggplot(data=condo_resale.sf, aes(x= `PROX_ELDERLYCARE`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_URA_GROWTH_AREA <- ggplot(data=condo_resale.sf, aes(x= `PROX_URA_GROWTH_AREA`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_HAWKER_MARKET <- ggplot(data=condo_resale.sf, aes(x= `PROX_HAWKER_MARKET`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_KINDERGARTEN <- ggplot(data=condo_resale.sf, aes(x= `PROX_KINDERGARTEN`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_MRT <- ggplot(data=condo_resale.sf, aes(x= `PROX_MRT`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_PARK <- ggplot(data=condo_resale.sf, aes(x= `PROX_PARK`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_PRIMARY_SCH <- ggplot(data=condo_resale.sf, aes(x= `PROX_PRIMARY_SCH`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_TOP_PRIMARY_SCH <- ggplot(data=condo_resale.sf, aes(x= `PROX_TOP_PRIMARY_SCH`)) +
geom_histogram(bins=20, color="black", fill="light blue")
ggarrange(AREA_SQM, AGE, PROX_CBD, PROX_CHILDCARE, PROX_ELDERLYCARE, PROX_URA_GROWTH_AREA, PROX_HAWKER_MARKET, PROX_KINDERGARTEN, PROX_MRT, PROX_PARK, PROX_PRIMARY_SCH, PROX_TOP_PRIMARY_SCH, ncol = 3, nrow = 4)
Most of the distribution seems to be relatively right skewed, however they do have resemblances to a gaussian distribution as well.
Drawing Statistical Point Map
Setting the plot mode to “view” for interactive viewing so that we can explore the spatial points at different areas of the map clearly.
tmap_mode("view")
tmap_options(check.and.fix = TRUE)+ # the sf package have a little bit of problem since their upgrade to s3, tmap mauy not understand the model well and flags out error msg. To avoid it, keep the sf package at lower version, 0.9-8 would be a good version to use.
tm_shape(mpsz_svy21)+
tm_polygons() +
tm_shape(condo_resale.sf) +
tm_dots(col = "SELLING_PRICE",
alpha = 0.6,
style="quantile") +
tm_view(set.zoom.limits = c(11,14))
Setting back the plot mode to static “plot” mode to prevent unnecessary calls.
tmap_mode("plot")
6. Hedonic Pricing Modelling in R
We will be using R base’s method, lm(), to build hedonic pricing models for the condo resale units.
Simple Linear Regression Model
Building a simple linear regression model using SELLING_PRICE as the dependent variable, or y variable, and AREA_SQM as the indendent variable, or x variable.
Thus forming a simple linear equation of SELLING_PRICE = B0 + B1*AREA_SQM + E, where B0 is the y-intercept, B1 is the degree of change to the y variable given 1 unit change of x variable, AREA_SQM, and lastly, E is the residual.
condo.slr <- lm(formula=SELLING_PRICE ~ AREA_SQM, data = condo_resale.sf)
Getting the values for B0, B1.
summary(condo.slr)
Call:
lm(formula = SELLING_PRICE ~ AREA_SQM, data = condo_resale.sf)
Residuals:
Min 1Q Median 3Q Max
-3695815 -391764 -87517 258900 13503875
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -258121.1 63517.2 -4.064 5.09e-05 ***
AREA_SQM 14719.0 428.1 34.381 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 942700 on 1434 degrees of freedom
Multiple R-squared: 0.4518, Adjusted R-squared: 0.4515
F-statistic: 1182 on 1 and 1434 DF, p-value: < 2.2e-16Analysis of variance:
anova(condo.slr)
Analysis of Variance Table
Response: SELLING_PRICE
Df Sum Sq Mean Sq F value Pr(>F)
AREA_SQM 1 1.0504e+15 1.0504e+15 1182 < 2.2e-16 ***
Residuals 1434 1.2743e+15 8.8861e+11
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Based on the summary stats, we can see that the y-intercept would be -258121.1, while the coefficient of AREA_SQM is 14719.0, thus giving us the relational formula between the two variable as:
SELLING_PRICE = -258121.1 + 14719.0 * AREA_SQM
We also know that the R-squared value is 0.4518, while adjusted R-square is 0.4515, both of which are rather low, signifying that roughly only 45% of the data can be explained by this regression model.
However, the hypothesis testing with p-value much lower than 0.0001 suggest that we can confidently reject the null hypothesis that mean is a good estimator of SELLING_PRICE and that the simple linear regression model above is a good estimator for SELLING_PRICE.
Both the coefficients have a p-value less than 0.0001 as well, thus we can confidently reject the null hypothesis that B0 and B1 are equal to 0, meaning that both B0 and B1 are good parameter estimates.
Visualizing the best fit curve
ggplot(data=condo_resale.sf,
aes(x=`AREA_SQM`, y=`SELLING_PRICE`)) +
geom_point() +
geom_smooth(method = lm)
Next, we will take a look at a more realistic regression model using more independent variables.
Multiple Linear Regression Model
Visualizing the relationships of the independent variable
Before we can build a multi-linear regression model, we need to ensure that the independent variables are not highly correlated to each other, as that will mean that the change in value to one highly correlated variable might affect the independent variable but it will also affect the other highly correlated variables.
To check for such correlation, or multicollinearity, a correlation matrix is commonly used to visualize these correlations.
corrplot(cor(condo_resale[, 5:23]), diag = FALSE, order = "AOE",
tl.pos = "td", tl.cex = 0.5, method = "number", type = "upper")
Note: diag = FALSE is used to show only one side of the correlation matrix, since the other half is the same.
Matrix reorder is very important for mining the hidden structure and patterns in the matrix. Four methods of matrix reorder in corrplot are:
- AOE
- FPC
- hclust
- alphabet
At a glance, we can see that Freehold is highly correlated to LEASE_99YEAR. In view of this, it is wiser to only include either one of them in the subsequent model building. As a result, LEASE_99YEAR is excluded in the subsequent model building.
Building a hedonic pricing model using multiple linear regression model
condo.mlr <- lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + PROX_HAWKER_MARKET + PROX_KINDERGARTEN + PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + PROX_TOP_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_SUPERMARKET + PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, data=condo_resale.sf)
summary(condo.mlr)
Call:
lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + PROX_CHILDCARE +
PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + PROX_HAWKER_MARKET +
PROX_KINDERGARTEN + PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH +
PROX_TOP_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_SUPERMARKET +
PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD,
data = condo_resale.sf)
Residuals:
Min 1Q Median 3Q Max
-3475964 -293923 -23069 241043 12260381
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 481728.40 121441.01 3.967 7.65e-05 ***
AREA_SQM 12708.32 369.59 34.385 < 2e-16 ***
AGE -24440.82 2763.16 -8.845 < 2e-16 ***
PROX_CBD -78669.78 6768.97 -11.622 < 2e-16 ***
PROX_CHILDCARE -351617.91 109467.25 -3.212 0.00135 **
PROX_ELDERLYCARE 171029.42 42110.51 4.061 5.14e-05 ***
PROX_URA_GROWTH_AREA 38474.53 12523.57 3.072 0.00217 **
PROX_HAWKER_MARKET 23746.10 29299.76 0.810 0.41782
PROX_KINDERGARTEN 147468.99 82668.87 1.784 0.07466 .
PROX_MRT -314599.68 57947.44 -5.429 6.66e-08 ***
PROX_PARK 563280.50 66551.68 8.464 < 2e-16 ***
PROX_PRIMARY_SCH 180186.08 65237.95 2.762 0.00582 **
PROX_TOP_PRIMARY_SCH 2280.04 20410.43 0.112 0.91107
PROX_SHOPPING_MALL -206604.06 42840.60 -4.823 1.57e-06 ***
PROX_SUPERMARKET -44991.80 77082.64 -0.584 0.55953
PROX_BUS_STOP 683121.35 138353.28 4.938 8.85e-07 ***
NO_Of_UNITS -231.18 89.03 -2.597 0.00951 **
FAMILY_FRIENDLY 140340.77 47020.55 2.985 0.00289 **
FREEHOLD 359913.01 49220.22 7.312 4.38e-13 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 755800 on 1417 degrees of freedom
Multiple R-squared: 0.6518, Adjusted R-squared: 0.6474
F-statistic: 147.4 on 18 and 1417 DF, p-value: < 2.2e-16From the statistical summary, we can see that not all variables are statistically significant, thus we will need to revise the model by removing these statistically insignificant variables. Namely: * PROX_HAWKER_MARKET * PROX_KINDERGARTEN * PROX_TOP_PRIMARY_SCH * PROX_SUPERMARKET
condo.mlr1 <- lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, data=condo_resale.sf)
summary(condo.mlr1)
Call:
lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + PROX_CHILDCARE +
PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + PROX_MRT + PROX_PARK +
PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_BUS_STOP + NO_Of_UNITS +
FAMILY_FRIENDLY + FREEHOLD, data = condo_resale.sf)
Residuals:
Min 1Q Median 3Q Max
-3470778 -298119 -23481 248917 12234210
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 527633.22 108183.22 4.877 1.20e-06 ***
AREA_SQM 12777.52 367.48 34.771 < 2e-16 ***
AGE -24687.74 2754.84 -8.962 < 2e-16 ***
PROX_CBD -77131.32 5763.12 -13.384 < 2e-16 ***
PROX_CHILDCARE -318472.75 107959.51 -2.950 0.003231 **
PROX_ELDERLYCARE 185575.62 39901.86 4.651 3.61e-06 ***
PROX_URA_GROWTH_AREA 39163.25 11754.83 3.332 0.000885 ***
PROX_MRT -294745.11 56916.37 -5.179 2.56e-07 ***
PROX_PARK 570504.81 65507.03 8.709 < 2e-16 ***
PROX_PRIMARY_SCH 159856.14 60234.60 2.654 0.008046 **
PROX_SHOPPING_MALL -220947.25 36561.83 -6.043 1.93e-09 ***
PROX_BUS_STOP 682482.22 134513.24 5.074 4.42e-07 ***
NO_Of_UNITS -245.48 87.95 -2.791 0.005321 **
FAMILY_FRIENDLY 146307.58 46893.02 3.120 0.001845 **
FREEHOLD 350599.81 48506.48 7.228 7.98e-13 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 756000 on 1421 degrees of freedom
Multiple R-squared: 0.6507, Adjusted R-squared: 0.6472
F-statistic: 189.1 on 14 and 1421 DF, p-value: < 2.2e-16Assumptions of linear regression models
Linearity Assumption: Relationship between dependent and independent variables is (approximately) linear.
Normality Assumption: The residual errors are assumed to be normally distributed.
Homogenuity of residual variance: The residuals are assumed to have a constant variance (homoscedasticity).
Residuals are independent to each other
(Optional) Errors are normally distributed with a population mean of 0.
Checking for multicollinearity
We will explore the use of olsrr, a package specially programmed for performing Ordinary Least Squared (OLS) regression.
Some of the useful methods includes: * comprehensive regression output * residual diagnostics * measures of influence * heteroskedasticity tests * collinearity diagnostics * model fit assessment * variable contribution assessment * variable selection procedures
We will be using ols_vid_tol() method from olsrr package for multicollinearity.
ols_vif_tol(condo.mlr1)
Variables Tolerance VIF
1 AREA_SQM 0.8728554 1.145665
2 AGE 0.7071275 1.414172
3 PROX_CBD 0.6356147 1.573280
4 PROX_CHILDCARE 0.3066019 3.261559
5 PROX_ELDERLYCARE 0.6598479 1.515501
6 PROX_URA_GROWTH_AREA 0.7510311 1.331503
7 PROX_MRT 0.5236090 1.909822
8 PROX_PARK 0.8279261 1.207837
9 PROX_PRIMARY_SCH 0.4524628 2.210126
10 PROX_SHOPPING_MALL 0.6738795 1.483945
11 PROX_BUS_STOP 0.3514118 2.845664
12 NO_Of_UNITS 0.6901036 1.449058
13 FAMILY_FRIENDLY 0.7244157 1.380423
14 FREEHOLD 0.6931163 1.442759A good judgement of multicollinearity would be if the VIF is above 10. Since none of the variables exceed the VIF value of 10, we can safely conclude that there are no sign of multicollinearity among the independent variables.
Testing for non-linearity
In multiple linear regression, we need to test for linearity and additivity of the relationship between dependent and independent variables. We can do so using ols_plot_resid_fit() from olsrr package to perform linearity assumption test.
ols_plot_resid_fit(condo.mlr1)
From the figure above, we can see that the residual roughly revolves around the 0 line, thus we can safely conclude that the relationships between the dependent and the independent variables are linear.
Testing for normality assumption
Next, we still need to test if the residual errors are normally distributed using ols_plot_resid_hist() of olsrr package to perform normality assumption test.
ols_plot_resid_hist(condo.mlr1)
The figure above shows that the residual of the multiple linear regression model does resemble a normal distribution.
A statistical approached introduced is ols_test-normality() of olsrr package as well:
ols_test_normality(condo.mlr1)
-----------------------------------------------
Test Statistic pvalue
-----------------------------------------------
Shapiro-Wilk 0.6856 0.0000
Kolmogorov-Smirnov 0.1366 0.0000
Cramer-von Mises 121.0768 0.0000
Anderson-Darling 67.9551 0.0000
-----------------------------------------------The summary table above reveals that the p-values of the four tests are way smaller than the alpha value of 0.05. Hence we will reject the null hypothesis that the residual is NOT resemble normal distribution.
Testing for Spatial Autocorrelation
The hedonic model we are building are using geographically referenced attribute, thus we should visualize the residual of the hedonic pricing model.
To perform spatial autocorrelation test, we will have to convert condo_resale.sf into SpatialPointDataFrame.
mlr.output <- as.data.frame(condo.mlr1$residuals)
Joining the newly created data frame with condo_resale.sf
condo_resale.res.sf <- cbind(condo_resale.sf,
condo.mlr1$residuals) %>%
rename(`MLR_RES` = `condo.mlr1.residuals`)
Converting the sf object into SpatialPointDataFrame using spdep package:
condo_resale.sp <- as_Spatial(condo_resale.res.sf)
condo_resale.sp
class : SpatialPointsDataFrame
features : 1436
extent : 14940.85, 43352.45, 24765.67, 48382.81 (xmin, xmax, ymin, ymax)
crs : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs
variables : 23
names : POSTCODE, SELLING_PRICE, AREA_SQM, AGE, PROX_CBD, PROX_CHILDCARE, PROX_ELDERLYCARE, PROX_URA_GROWTH_AREA, PROX_HAWKER_MARKET, PROX_KINDERGARTEN, PROX_MRT, PROX_PARK, PROX_PRIMARY_SCH, PROX_TOP_PRIMARY_SCH, PROX_SHOPPING_MALL, ...
min values : 18965, 540000, 34, 0, 0.386916393, 0.004927023, 0.054508623, 0.214539508, 0.051817113, 0.004927023, 0.052779424, 0.029064164, 0.077106132, 0.077106132, 0, ...
max values : 828833, 1.8e+07, 619, 37, 19.18042832, 3.46572633, 3.949157205, 9.15540001, 5.374348075, 2.229045366, 3.48037319, 2.16104919, 3.928989144, 6.748192062, 3.477433767, ... Now we can plot a interactive visualization of the residual on a map itself.
First, setting the tmap mode to “view”, or interactive.
tmap_mode("view")
Plotting the geographically referenced residual:
tm_shape(mpsz_svy21)+
tm_polygons(alpha = 0.4) +
tm_shape(condo_resale.res.sf) +
tm_dots(col = "MLR_RES",
alpha = 0.6,
style="quantile") +
tm_view(set.zoom.limits = c(11,14))
Setting back the tmap mode to “plot”:
tmap_mode("plot")
The above plot does show signs of spatial autocorrelation, however, to be more definitive, we will use Moran’s I test to confirm our observation.
First, computing the distance-based weight matrix using dnearneigh() function of spdep:
nb <- dnearneigh(coordinates(condo_resale.sp), 0, 1500, longlat = FALSE)
summary(nb)
Neighbour list object:
Number of regions: 1436
Number of nonzero links: 66266
Percentage nonzero weights: 3.213526
Average number of links: 46.14624
Link number distribution:
1 3 5 7 9 10 11 12 13 14 15 16 17 18 19 20 21
3 3 9 4 3 15 10 19 17 45 19 5 14 29 19 6 35
22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
45 18 47 16 43 22 26 21 11 9 23 22 13 16 25 21 37
39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55
16 18 8 21 4 12 8 36 18 14 14 43 11 12 8 13 12
56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72
13 4 5 6 12 11 20 29 33 15 20 10 14 15 15 11 16
73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89
12 10 8 19 12 14 9 8 4 13 11 6 4 9 4 4 4
90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106
6 2 16 9 4 5 9 3 9 4 2 1 2 1 1 1 5
107 108 109 110 112 116 125
9 2 1 3 1 1 1
3 least connected regions:
193 194 277 with 1 link
1 most connected region:
285 with 125 linksNext, nb2listw() of spdep packge will be used to convert the output neighbours lists (i.e. nb) into a spatial weights.
nb_lw <- nb2listw(nb, style = 'W')
summary(nb_lw)
Characteristics of weights list object:
Neighbour list object:
Number of regions: 1436
Number of nonzero links: 66266
Percentage nonzero weights: 3.213526
Average number of links: 46.14624
Link number distribution:
1 3 5 7 9 10 11 12 13 14 15 16 17 18 19 20 21
3 3 9 4 3 15 10 19 17 45 19 5 14 29 19 6 35
22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
45 18 47 16 43 22 26 21 11 9 23 22 13 16 25 21 37
39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55
16 18 8 21 4 12 8 36 18 14 14 43 11 12 8 13 12
56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72
13 4 5 6 12 11 20 29 33 15 20 10 14 15 15 11 16
73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89
12 10 8 19 12 14 9 8 4 13 11 6 4 9 4 4 4
90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106
6 2 16 9 4 5 9 3 9 4 2 1 2 1 1 1 5
107 108 109 110 112 116 125
9 2 1 3 1 1 1
3 least connected regions:
193 194 277 with 1 link
1 most connected region:
285 with 125 links
Weights style: W
Weights constants summary:
n nn S0 S1 S2
W 1436 2062096 1436 94.81916 5798.341Next, lm.morantest() of spdep package will be used to perform Moran’s I test for residual spatial autocorrelation
lm.morantest(condo.mlr1, nb_lw)
Global Moran I for regression residuals
data:
model: lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD
+ PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA +
PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL +
PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, data
= condo_resale.sf)
weights: nb_lw
Moran I statistic standard deviate = 24.366, p-value < 2.2e-16
alternative hypothesis: greater
sample estimates:
Observed Moran I Expectation Variance
1.438876e-01 -5.487594e-03 3.758259e-05 Based on the global moran’s I test, the residual spatial autocorrelation shows that it’s p-value is less than 2.2 x 10^-16, which is a significantly small value, and is much lower than the alpha value of 0.05, hence we will reject the null hypothesis that the residuals are randomly distribute, in other words, the residuals resembles cluster distributions.
7. Building Hedonic Pricing Models using GWmodel
In this section, we will learn how to model hedonic pricing using both the fixed and adaptive bandwidth schemes
Building Fixed Bandwidth GWR Model
Computing Fixed Bandwidth
Firstly, we will be using bw.gwr() of the GWmodel package to determine the optimal fixed bandwidth to use in the model, we will set the argument, adaptive, to FALSE since we will explore fixed bandwidth first.
There are two approaches to determine the stopping rule. 1. CV cross-validation approach 2. AIC corrected approach
bw.fixed <- bw.gwr(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, data=condo_resale.sp, approach="CV", kernel="gaussian", adaptive=FALSE, longlat=FALSE)
Fixed bandwidth: 17660.96 CV score: 8.259118e+14
Fixed bandwidth: 10917.26 CV score: 7.970454e+14
Fixed bandwidth: 6749.419 CV score: 7.273273e+14
Fixed bandwidth: 4173.553 CV score: 6.300006e+14
Fixed bandwidth: 2581.58 CV score: 5.404958e+14
Fixed bandwidth: 1597.687 CV score: 4.857515e+14
Fixed bandwidth: 989.6077 CV score: 4.722431e+14
Fixed bandwidth: 613.7939 CV score: 1.378294e+16
Fixed bandwidth: 1221.873 CV score: 4.778717e+14
Fixed bandwidth: 846.0596 CV score: 4.791629e+14
Fixed bandwidth: 1078.325 CV score: 4.751406e+14
Fixed bandwidth: 934.7772 CV score: 4.72518e+14
Fixed bandwidth: 1023.495 CV score: 4.730305e+14
Fixed bandwidth: 968.6643 CV score: 4.721317e+14
Fixed bandwidth: 955.7206 CV score: 4.722072e+14
Fixed bandwidth: 976.6639 CV score: 4.721387e+14
Fixed bandwidth: 963.7202 CV score: 4.721484e+14
Fixed bandwidth: 971.7199 CV score: 4.721293e+14
Fixed bandwidth: 973.6083 CV score: 4.721309e+14
Fixed bandwidth: 970.5527 CV score: 4.721295e+14
Fixed bandwidth: 972.4412 CV score: 4.721296e+14
Fixed bandwidth: 971.2741 CV score: 4.721292e+14
Fixed bandwidth: 970.9985 CV score: 4.721293e+14
Fixed bandwidth: 971.4443 CV score: 4.721292e+14
Fixed bandwidth: 971.5496 CV score: 4.721293e+14
Fixed bandwidth: 971.3793 CV score: 4.721292e+14
Fixed bandwidth: 971.3391 CV score: 4.721292e+14
Fixed bandwidth: 971.3143 CV score: 4.721292e+14
Fixed bandwidth: 971.3545 CV score: 4.721292e+14
Fixed bandwidth: 971.3296 CV score: 4.721292e+14
Fixed bandwidth: 971.345 CV score: 4.721292e+14
Fixed bandwidth: 971.3355 CV score: 4.721292e+14
Fixed bandwidth: 971.3413 CV score: 4.721292e+14
Fixed bandwidth: 971.3377 CV score: 4.721292e+14
Fixed bandwidth: 971.34 CV score: 4.721292e+14
Fixed bandwidth: 971.3405 CV score: 4.721292e+14
Fixed bandwidth: 971.3408 CV score: 4.721292e+14
Fixed bandwidth: 971.3403 CV score: 4.721292e+14
Fixed bandwidth: 971.3406 CV score: 4.721292e+14
Fixed bandwidth: 971.3404 CV score: 4.721292e+14
Fixed bandwidth: 971.3405 CV score: 4.721292e+14
Fixed bandwidth: 971.3405 CV score: 4.721292e+14 The result shows that the recommended bandwidth is 971.4443 metres. The reason why it is in metres is because the projection system for CRS 3414 is in metres.
GWmodel method - fixed bandwith
Now we can use the code chunk below to calibrate the gwr model using fixed bandwidth and gaussian kernel.
gwr.fixed <- gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, data=condo_resale.sp, bw=bw.fixed, kernel = 'gaussian', longlat = FALSE)
The output is saved in a list of class “gwrm”. The code below can be used to display the model output.
gwr.fixed
***********************************************************************
* Package GWmodel *
***********************************************************************
Program starts at: 2021-10-18 11:41:12
Call:
gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD +
PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA +
PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL +
PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD,
data = condo_resale.sp, bw = bw.fixed, kernel = "gaussian",
longlat = FALSE)
Dependent (y) variable: SELLING_PRICE
Independent variables: AREA_SQM AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE PROX_URA_GROWTH_AREA PROX_MRT PROX_PARK PROX_PRIMARY_SCH PROX_SHOPPING_MALL PROX_BUS_STOP NO_Of_UNITS FAMILY_FRIENDLY FREEHOLD
Number of data points: 1436
***********************************************************************
* Results of Global Regression *
***********************************************************************
Call:
lm(formula = formula, data = data)
Residuals:
Min 1Q Median 3Q Max
-3470778 -298119 -23481 248917 12234210
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 527633.22 108183.22 4.877 1.20e-06 ***
AREA_SQM 12777.52 367.48 34.771 < 2e-16 ***
AGE -24687.74 2754.84 -8.962 < 2e-16 ***
PROX_CBD -77131.32 5763.12 -13.384 < 2e-16 ***
PROX_CHILDCARE -318472.75 107959.51 -2.950 0.003231 **
PROX_ELDERLYCARE 185575.62 39901.86 4.651 3.61e-06 ***
PROX_URA_GROWTH_AREA 39163.25 11754.83 3.332 0.000885 ***
PROX_MRT -294745.11 56916.37 -5.179 2.56e-07 ***
PROX_PARK 570504.81 65507.03 8.709 < 2e-16 ***
PROX_PRIMARY_SCH 159856.14 60234.60 2.654 0.008046 **
PROX_SHOPPING_MALL -220947.25 36561.83 -6.043 1.93e-09 ***
PROX_BUS_STOP 682482.22 134513.24 5.074 4.42e-07 ***
NO_Of_UNITS -245.48 87.95 -2.791 0.005321 **
FAMILY_FRIENDLY 146307.58 46893.02 3.120 0.001845 **
FREEHOLD 350599.81 48506.48 7.228 7.98e-13 ***
---Significance stars
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 756000 on 1421 degrees of freedom
Multiple R-squared: 0.6507
Adjusted R-squared: 0.6472
F-statistic: 189.1 on 14 and 1421 DF, p-value: < 2.2e-16
***Extra Diagnostic information
Residual sum of squares: 8.120609e+14
Sigma(hat): 752522.9
AIC: 42966.76
AICc: 42967.14
BIC: 41731.39
***********************************************************************
* Results of Geographically Weighted Regression *
***********************************************************************
*********************Model calibration information*********************
Kernel function: gaussian
Fixed bandwidth: 971.3405
Regression points: the same locations as observations are used.
Distance metric: Euclidean distance metric is used.
****************Summary of GWR coefficient estimates:******************
Min. 1st Qu. Median
Intercept -3.5988e+07 -5.1998e+05 7.6780e+05
AREA_SQM 1.0003e+03 5.2758e+03 7.4740e+03
AGE -1.3475e+05 -2.0813e+04 -8.6260e+03
PROX_CBD -7.7047e+07 -2.3608e+05 -8.3600e+04
PROX_CHILDCARE -6.0097e+06 -3.3667e+05 -9.7425e+04
PROX_ELDERLYCARE -3.5000e+06 -1.5970e+05 3.1971e+04
PROX_URA_GROWTH_AREA -3.0170e+06 -8.2013e+04 7.0749e+04
PROX_MRT -3.5282e+06 -6.5836e+05 -1.8833e+05
PROX_PARK -1.2062e+06 -2.1732e+05 3.5383e+04
PROX_PRIMARY_SCH -2.2695e+07 -1.7066e+05 4.8472e+04
PROX_SHOPPING_MALL -7.2585e+06 -1.6684e+05 -1.0517e+04
PROX_BUS_STOP -1.4676e+06 -4.5207e+04 3.7601e+05
NO_Of_UNITS -1.3170e+03 -2.4822e+02 -3.0846e+01
FAMILY_FRIENDLY -2.2749e+06 -1.1140e+05 7.6214e+03
FREEHOLD -9.2067e+06 3.8073e+04 1.5169e+05
3rd Qu. Max.
Intercept 1.7412e+06 112793548
AREA_SQM 1.2301e+04 21575
AGE -3.7784e+03 434201
PROX_CBD 3.4646e+04 2704596
PROX_CHILDCARE 2.9007e+05 1654087
PROX_ELDERLYCARE 1.9577e+05 38867814
PROX_URA_GROWTH_AREA 2.2612e+05 78515730
PROX_MRT 3.6922e+04 3124316
PROX_PARK 4.1335e+05 18122425
PROX_PRIMARY_SCH 5.1555e+05 4637503
PROX_SHOPPING_MALL 1.5923e+05 1529952
PROX_BUS_STOP 1.1664e+06 11342182
NO_Of_UNITS 2.5496e+02 12907
FAMILY_FRIENDLY 1.6107e+05 1720744
FREEHOLD 3.7528e+05 6073636
************************Diagnostic information*************************
Number of data points: 1436
Effective number of parameters (2trace(S) - trace(S'S)): 438.3804
Effective degrees of freedom (n-2trace(S) + trace(S'S)): 997.6196
AICc (GWR book, Fotheringham, et al. 2002, p. 61, eq 2.33): 42263.61
AIC (GWR book, Fotheringham, et al. 2002,GWR p. 96, eq. 4.22): 41632.36
BIC (GWR book, Fotheringham, et al. 2002,GWR p. 61, eq. 2.34): 42515.71
Residual sum of squares: 2.53407e+14
R-square value: 0.8909912
Adjusted R-square value: 0.8430417
***********************************************************************
Program stops at: 2021-10-18 11:41:13 The report shows that the adjusted r-square of the gwr is 0.8430418 which is significantly better than the globel multiple linear regression model of 0.6472.
Now, let’s move onto explore Adaptive bandwidth GWR modelling instead.
Building Adaptive Bandwidth GWR Model
In this section, we will calibrate the gwr-absed hedonic pricing model by using adaptive bandwidth approach.
Computing the adaptive bandwidth
We will be using bw.ger() to determine the recommended data point to use.
The code chunk used look very similar to the one used to compute the fixed bandwidth except the adaptive argument has changed to TRUE.
bw.adaptive <- bw.gwr(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, data=condo_resale.sp, approach="CV", kernel="gaussian",
adaptive=TRUE, longlat=FALSE)
Adaptive bandwidth: 895 CV score: 7.952401e+14
Adaptive bandwidth: 561 CV score: 7.667364e+14
Adaptive bandwidth: 354 CV score: 6.953454e+14
Adaptive bandwidth: 226 CV score: 6.15223e+14
Adaptive bandwidth: 147 CV score: 5.674373e+14
Adaptive bandwidth: 98 CV score: 5.426745e+14
Adaptive bandwidth: 68 CV score: 5.168117e+14
Adaptive bandwidth: 49 CV score: 4.859631e+14
Adaptive bandwidth: 37 CV score: 4.646518e+14
Adaptive bandwidth: 30 CV score: 4.422088e+14
Adaptive bandwidth: 25 CV score: 4.430816e+14
Adaptive bandwidth: 32 CV score: 4.505602e+14
Adaptive bandwidth: 27 CV score: 4.462172e+14
Adaptive bandwidth: 30 CV score: 4.422088e+14 The result shows that the 30 is the recommended data points to be used.
Constructing the adaptive bandwidth gwr model
Now, we can calibrate the gwr-based hedonic pricing model by using adaptive bandwidth and gaussian kernel.
gwr.adaptive <- gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, data=condo_resale.sp, bw=bw.adaptive, kernel = 'gaussian', adaptive=TRUE, longlat = FALSE)
gwr.adaptive
***********************************************************************
* Package GWmodel *
***********************************************************************
Program starts at: 2021-10-18 11:41:21
Call:
gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD +
PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA +
PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL +
PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD,
data = condo_resale.sp, bw = bw.adaptive, kernel = "gaussian",
adaptive = TRUE, longlat = FALSE)
Dependent (y) variable: SELLING_PRICE
Independent variables: AREA_SQM AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE PROX_URA_GROWTH_AREA PROX_MRT PROX_PARK PROX_PRIMARY_SCH PROX_SHOPPING_MALL PROX_BUS_STOP NO_Of_UNITS FAMILY_FRIENDLY FREEHOLD
Number of data points: 1436
***********************************************************************
* Results of Global Regression *
***********************************************************************
Call:
lm(formula = formula, data = data)
Residuals:
Min 1Q Median 3Q Max
-3470778 -298119 -23481 248917 12234210
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 527633.22 108183.22 4.877 1.20e-06 ***
AREA_SQM 12777.52 367.48 34.771 < 2e-16 ***
AGE -24687.74 2754.84 -8.962 < 2e-16 ***
PROX_CBD -77131.32 5763.12 -13.384 < 2e-16 ***
PROX_CHILDCARE -318472.75 107959.51 -2.950 0.003231 **
PROX_ELDERLYCARE 185575.62 39901.86 4.651 3.61e-06 ***
PROX_URA_GROWTH_AREA 39163.25 11754.83 3.332 0.000885 ***
PROX_MRT -294745.11 56916.37 -5.179 2.56e-07 ***
PROX_PARK 570504.81 65507.03 8.709 < 2e-16 ***
PROX_PRIMARY_SCH 159856.14 60234.60 2.654 0.008046 **
PROX_SHOPPING_MALL -220947.25 36561.83 -6.043 1.93e-09 ***
PROX_BUS_STOP 682482.22 134513.24 5.074 4.42e-07 ***
NO_Of_UNITS -245.48 87.95 -2.791 0.005321 **
FAMILY_FRIENDLY 146307.58 46893.02 3.120 0.001845 **
FREEHOLD 350599.81 48506.48 7.228 7.98e-13 ***
---Significance stars
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 756000 on 1421 degrees of freedom
Multiple R-squared: 0.6507
Adjusted R-squared: 0.6472
F-statistic: 189.1 on 14 and 1421 DF, p-value: < 2.2e-16
***Extra Diagnostic information
Residual sum of squares: 8.120609e+14
Sigma(hat): 752522.9
AIC: 42966.76
AICc: 42967.14
BIC: 41731.39
***********************************************************************
* Results of Geographically Weighted Regression *
***********************************************************************
*********************Model calibration information*********************
Kernel function: gaussian
Adaptive bandwidth: 30 (number of nearest neighbours)
Regression points: the same locations as observations are used.
Distance metric: Euclidean distance metric is used.
****************Summary of GWR coefficient estimates:******************
Min. 1st Qu. Median
Intercept -1.3487e+08 -2.4669e+05 7.7928e+05
AREA_SQM 3.3188e+03 5.6285e+03 7.7825e+03
AGE -9.6746e+04 -2.9288e+04 -1.4043e+04
PROX_CBD -2.5330e+06 -1.6256e+05 -7.7242e+04
PROX_CHILDCARE -1.2790e+06 -2.0175e+05 8.7158e+03
PROX_ELDERLYCARE -1.6212e+06 -9.2050e+04 6.1029e+04
PROX_URA_GROWTH_AREA -7.2686e+06 -3.0350e+04 4.5869e+04
PROX_MRT -4.3781e+07 -6.7282e+05 -2.2115e+05
PROX_PARK -2.9020e+06 -1.6782e+05 1.1601e+05
PROX_PRIMARY_SCH -8.6418e+05 -1.6627e+05 -7.7853e+03
PROX_SHOPPING_MALL -1.8272e+06 -1.3175e+05 -1.4049e+04
PROX_BUS_STOP -2.0579e+06 -7.1461e+04 4.1104e+05
NO_Of_UNITS -2.1993e+03 -2.3685e+02 -3.4699e+01
FAMILY_FRIENDLY -5.9879e+05 -5.0927e+04 2.6173e+04
FREEHOLD -1.6340e+05 4.0765e+04 1.9023e+05
3rd Qu. Max.
Intercept 1.6194e+06 18758355
AREA_SQM 1.2738e+04 23064
AGE -5.6119e+03 13303
PROX_CBD 2.6624e+03 11346650
PROX_CHILDCARE 3.7778e+05 2892127
PROX_ELDERLYCARE 2.8184e+05 2465671
PROX_URA_GROWTH_AREA 2.4613e+05 7384059
PROX_MRT -7.4593e+04 1186242
PROX_PARK 4.6572e+05 2588497
PROX_PRIMARY_SCH 4.3222e+05 3381462
PROX_SHOPPING_MALL 1.3799e+05 38038564
PROX_BUS_STOP 1.2071e+06 12081592
NO_Of_UNITS 1.1657e+02 1010
FAMILY_FRIENDLY 2.2481e+05 2072414
FREEHOLD 3.7960e+05 1813995
************************Diagnostic information*************************
Number of data points: 1436
Effective number of parameters (2trace(S) - trace(S'S)): 350.3088
Effective degrees of freedom (n-2trace(S) + trace(S'S)): 1085.691
AICc (GWR book, Fotheringham, et al. 2002, p. 61, eq 2.33): 41982.22
AIC (GWR book, Fotheringham, et al. 2002,GWR p. 96, eq. 4.22): 41546.74
BIC (GWR book, Fotheringham, et al. 2002,GWR p. 61, eq. 2.34): 41914.08
Residual sum of squares: 2.528227e+14
R-square value: 0.8912425
Adjusted R-square value: 0.8561185
***********************************************************************
Program stops at: 2021-10-18 11:41:23 The report shows that the adjusted r-square of the gwr is 0.8561185 which is significantly better than the globel multiple linear regression model of 0.6472.
8. Visualizing the GWR output
Prof Notes:
In addition to regression residuals, the output feature class table includes fields for observed and predicted y values, condition number (cond), Local R2, residuals, and explanatory variable coefficients and standard errors:
Condition Number: this diagnostic evaluates local collinearity. In the presence of strong local collinearity, results become unstable. Results associated with condition numbers larger than 30, may be unreliable.
Local R2: these values range between 0.0 and 1.0 and indicate how well the local regression model fits observed y values. Very low values indicate the local model is performing poorly. Mapping the Local R2 values to see where GWR predicts well and where it predicts poorly may provide clues about important variables that may be missing from the regression model.
Predicted: these are the estimated (or fitted) y values 3. computed by GWR.
Residuals: to obtain the residual values, the fitted y values are subtracted from the observed y values. Standardized residuals have a mean of zero and a standard deviation of 1. A cold-to-hot rendered map of standardized residuals can be produce by using these values.
Coefficient Standard Error: these values measure the reliability of each coefficient estimate. Confidence in those estimates are higher when standard errors are small in relation to the actual coefficient values. Large standard errors may indicate problems with local collinearity.
They are all stored in a SpatialPointsDataFrame or SpatialPolygonsDataFrame object integrated with fit.points, GWR coefficient estimates, y value, predicted values, coefficient standard errors and t-values in its “data” slot in an object called SDF of the output list.
Converting SDF into sf data.frame
To visualize the fields in SDF object, we need to convert the output into sf data.frame first:
condo_resale.sf.adaptive <- st_as_sf(gwr.adaptive$SDF) %>%
st_transform(crs=3414)
Setting the projection:
condo_resale.sf.adaptive.svy21 <- st_transform(condo_resale.sf.adaptive, 3414)
condo_resale.sf.adaptive.svy21
Simple feature collection with 1436 features and 51 fields
Geometry type: POINT
Dimension: XY
Bounding box: xmin: 14940.85 ymin: 24765.67 xmax: 43352.45 ymax: 48382.81
Projected CRS: SVY21 / Singapore TM
First 10 features:
Intercept AREA_SQM AGE PROX_CBD PROX_CHILDCARE
1 2050011.7 9561.892 -9514.634 -120681.9 319266.92
2 1633128.2 16576.853 -58185.479 -149434.2 441102.18
3 3433608.2 13091.861 -26707.386 -259397.8 -120116.82
4 234358.9 20730.601 -93308.988 2426853.7 480825.28
5 2285804.9 6722.836 -17608.018 -316835.5 90764.78
6 -3568877.4 6039.581 -26535.592 327306.1 -152531.19
7 -2874842.4 16843.575 -59166.727 -983577.2 -177810.50
8 2038086.0 6905.135 -17681.897 -285076.6 70259.40
9 1718478.4 9580.703 -14401.128 105803.4 -657698.02
10 3457054.0 14072.011 -31579.884 -234895.4 79961.45
PROX_ELDERLYCARE PROX_URA_GROWTH_AREA PROX_MRT PROX_PARK
1 -393417.79 -159980.20 -299742.96 -172104.47
2 325188.74 -142290.39 -2510522.23 523379.72
3 535855.81 -253621.21 -936853.28 209099.85
4 314783.72 -2679297.89 -2039479.50 -759153.26
5 -137384.61 303714.81 -44567.05 -10284.62
6 -700392.85 -28051.25 733566.47 1511488.92
7 -122384.02 1397676.38 -2745430.34 710114.74
8 -96012.78 269368.71 -14552.99 73533.34
9 -123276.00 -361974.72 -476785.32 -132067.59
10 548581.04 -150024.38 -1503835.53 574155.47
PROX_PRIMARY_SCH PROX_SHOPPING_MALL PROX_BUS_STOP NO_Of_UNITS
1 242668.03 300881.390 1210615.4 104.8290640
2 1106830.66 -87693.378 1843587.2 -288.3441183
3 571462.33 -126732.712 1411924.9 -9.5532945
4 3127477.21 -29593.342 7225577.5 -161.3551620
5 30413.56 -7490.586 677577.0 42.2659674
6 320878.23 258583.881 1086012.6 -214.3671271
7 1786570.95 -384251.210 5094060.5 -0.9212521
8 53359.73 -39634.902 735767.1 30.1741069
9 -40128.92 276718.757 2815772.4 675.1615559
10 108996.67 -454726.822 2123557.0 -21.3044311
FAMILY_FRIENDLY FREEHOLD y yhat residual CV_Score
1 -9075.370 303955.6 3000000 2886532 113468.16 0
2 310074.664 396221.3 3880000 3466801 413198.52 0
3 5949.746 168821.7 3325000 3616527 -291527.20 0
4 1556178.531 1212515.6 4250000 5435482 -1185481.63 0
5 58986.951 328175.2 1400000 1388166 11834.26 0
6 201992.641 471873.1 1320000 1516702 -196701.94 0
7 359659.512 408871.9 3410000 3266881 143118.77 0
8 55602.506 347075.0 1420000 1431955 -11955.27 0
9 -30453.297 503872.8 2025000 1832799 192200.83 0
10 -100935.586 213324.6 2550000 2223364 326635.53 0
Stud_residual Intercept_SE AREA_SQM_SE AGE_SE PROX_CBD_SE
1 0.38207013 516105.5 823.2860 5889.782 37411.22
2 1.01433140 488083.5 825.2380 6226.916 23615.06
3 -0.83780678 963711.4 988.2240 6510.236 56103.77
4 -2.84614670 444185.5 617.4007 6010.511 469337.41
5 0.03404453 2119620.6 1376.2778 8180.361 410644.47
6 -0.72065800 28572883.7 2348.0091 14601.909 5272846.47
7 0.41291992 679546.6 893.5893 8970.629 346164.20
8 -0.03033109 2217773.1 1415.2604 8661.309 438035.69
9 0.52018109 814281.8 943.8434 11791.208 89148.35
10 1.10559735 2410252.0 1271.4073 9941.980 173532.77
PROX_CHILDCARE_SE PROX_ELDERLYCARE_SE PROX_URA_GROWTH_AREA_SE
1 319111.1 120633.34 56207.39
2 299705.3 84546.69 76956.50
3 349128.5 129687.07 95774.60
4 304965.2 127150.69 470762.12
5 698720.6 327371.55 474339.56
6 1141599.8 1653002.19 5496627.21
7 530101.1 148598.71 371692.97
8 742532.8 399221.05 517977.91
9 704630.7 329683.30 153436.22
10 500976.2 281876.74 239182.57
PROX_MRT_SE PROX_PARK_SE PROX_PRIMARY_SCH_SE PROX_SHOPPING_MALL_SE
1 185181.3 205499.6 152400.7 109268.8
2 281133.9 229358.7 165150.7 98906.8
3 275483.7 314124.3 196662.6 119913.3
4 279877.1 227249.4 240878.9 177104.1
5 363830.0 364580.9 249087.7 301032.9
6 730453.2 1741712.0 683265.5 2931208.6
7 375511.9 297400.9 344602.8 249969.5
8 423155.4 440984.4 261251.2 351634.0
9 285325.4 304998.4 278258.5 289872.7
10 571355.7 599131.8 331284.8 265529.7
PROX_BUS_STOP_SE NO_Of_UNITS_SE FAMILY_FRIENDLY_SE FREEHOLD_SE
1 600668.6 218.1258 131474.7 115954.0
2 410222.1 208.9410 114989.1 130110.0
3 464156.7 210.9828 146607.2 141031.5
4 562810.8 361.7767 108726.6 138239.1
5 740922.4 299.5034 160663.7 210641.1
6 1418333.3 602.5571 331727.0 374347.3
7 821236.4 532.1978 129241.2 182216.9
8 775038.4 338.6777 171895.1 216649.4
9 850095.5 439.9037 220223.4 220473.7
10 631399.2 259.0169 189125.5 206346.2
Intercept_TV AREA_SQM_TV AGE_TV PROX_CBD_TV PROX_CHILDCARE_TV
1 3.9720784 11.614302 -1.615447 -3.22582173 1.00048819
2 3.3460017 20.087361 -9.344188 -6.32792021 1.47178634
3 3.5629010 13.247868 -4.102368 -4.62353528 -0.34404755
4 0.5276150 33.577223 -15.524302 5.17080808 1.57665606
5 1.0784029 4.884795 -2.152474 -0.77155660 0.12990138
6 -0.1249043 2.572214 -1.817269 0.06207388 -0.13361179
7 -4.2305303 18.849348 -6.595605 -2.84136028 -0.33542751
8 0.9189786 4.879056 -2.041481 -0.65080678 0.09462126
9 2.1104224 10.150733 -1.221345 1.18682383 -0.93339393
10 1.4343123 11.068059 -3.176418 -1.35360852 0.15961128
PROX_ELDERLYCARE_TV PROX_URA_GROWTH_AREA_TV PROX_MRT_TV
1 -3.2612693 -2.846248368 -1.61864578
2 3.8462625 -1.848971738 -8.92998600
3 4.1319138 -2.648105057 -3.40075727
4 2.4756745 -5.691404992 -7.28705261
5 -0.4196596 0.640289855 -0.12249416
6 -0.4237096 -0.005103357 1.00426206
7 -0.8235874 3.760298131 -7.31116712
8 -0.2405003 0.520038994 -0.03439159
9 -0.3739225 -2.359121712 -1.67102293
10 1.9461735 -0.627237944 -2.63204802
PROX_PARK_TV PROX_PRIMARY_SCH_TV PROX_SHOPPING_MALL_TV
1 -0.83749312 1.5923022 2.75358842
2 2.28192684 6.7019454 -0.88662640
3 0.66565951 2.9058009 -1.05686949
4 -3.34061770 12.9836105 -0.16709578
5 -0.02820944 0.1220998 -0.02488294
6 0.86781794 0.4696245 0.08821750
7 2.38773567 5.1844351 -1.53719231
8 0.16674816 0.2042469 -0.11271635
9 -0.43301073 -0.1442145 0.95462153
10 0.95831249 0.3290120 -1.71252687
PROX_BUS_STOP_TV NO_Of_UNITS_TV FAMILY_FRIENDLY_TV FREEHOLD_TV
1 2.0154464 0.480589953 -0.06902748 2.621347
2 4.4941192 -1.380026395 2.69655779 3.045280
3 3.0419145 -0.045279967 0.04058290 1.197050
4 12.8383775 -0.446007570 14.31276425 8.771149
5 0.9145046 0.141120178 0.36714544 1.557983
6 0.7656963 -0.355762335 0.60891234 1.260522
7 6.2029165 -0.001731033 2.78285441 2.243875
8 0.9493299 0.089093858 0.32346758 1.602012
9 3.3123012 1.534793921 -0.13828365 2.285410
10 3.3632555 -0.082251138 -0.53369623 1.033819
Local_R2 geometry
1 0.8846744 POINT (22085.12 29951.54)
2 0.8899773 POINT (25656.84 34546.2)
3 0.8947007 POINT (23963.99 32890.8)
4 0.9073605 POINT (27044.28 32319.77)
5 0.9510057 POINT (41042.56 33743.64)
6 0.9247586 POINT (39717.04 32943.1)
7 0.8310458 POINT (28419.1 33513.37)
8 0.9463936 POINT (40763.57 33879.61)
9 0.8380365 POINT (23595.63 28884.78)
10 0.9080753 POINT (24586.56 33194.31)gwr.adaptive.output <- as.data.frame(gwr.adaptive$SDF)
condo_resale.sf.adaptive <- cbind(condo_resale.res.sf, as.matrix(gwr.adaptive.output))
glimpse(condo_resale.sf.adaptive)
Rows: 1,436
Columns: 77
$ POSTCODE <dbl> 118635, 288420, 267833, 258380, 4671~
$ SELLING_PRICE <dbl> 3000000, 3880000, 3325000, 4250000, ~
$ AREA_SQM <dbl> 309, 290, 248, 127, 145, 139, 218, 1~
$ AGE <dbl> 30, 32, 33, 7, 28, 22, 24, 24, 27, 3~
$ PROX_CBD <dbl> 7.941259, 6.609797, 6.898000, 4.0388~
$ PROX_CHILDCARE <dbl> 0.16597932, 0.28027246, 0.42922669, ~
$ PROX_ELDERLYCARE <dbl> 2.5198118, 1.9333338, 0.5021395, 1.9~
$ PROX_URA_GROWTH_AREA <dbl> 6.618741, 7.505109, 6.463887, 4.9065~
$ PROX_HAWKER_MARKET <dbl> 1.76542207, 0.54507614, 0.37789301, ~
$ PROX_KINDERGARTEN <dbl> 0.05835552, 0.61592412, 0.14120309, ~
$ PROX_MRT <dbl> 0.5607188, 0.6584461, 0.3053433, 0.6~
$ PROX_PARK <dbl> 1.1710446, 0.1992269, 0.2779886, 0.9~
$ PROX_PRIMARY_SCH <dbl> 1.6340256, 0.9747834, 1.4715016, 1.4~
$ PROX_TOP_PRIMARY_SCH <dbl> 3.3273195, 0.9747834, 1.4715016, 2.3~
$ PROX_SHOPPING_MALL <dbl> 2.2102717, 2.9374279, 1.2256850, 0.3~
$ PROX_SUPERMARKET <dbl> 0.9103958, 0.5900617, 0.4135583, 0.4~
$ PROX_BUS_STOP <dbl> 0.10336166, 0.28673408, 0.28504777, ~
$ NO_Of_UNITS <dbl> 18, 20, 27, 30, 30, 31, 32, 32, 32, ~
$ FAMILY_FRIENDLY <dbl> 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, ~
$ FREEHOLD <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, ~
$ LEASEHOLD_99YR <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ~
$ LOG_SELLING_PRICE <dbl> 14.91412, 15.17135, 15.01698, 15.262~
$ MLR_RES <dbl> -1489099.55, 415494.57, 194129.69, 1~
$ Intercept <dbl> 2050011.67, 1633128.24, 3433608.17, ~
$ AREA_SQM.1 <dbl> 9561.892, 16576.853, 13091.861, 2073~
$ AGE.1 <dbl> -9514.634, -58185.479, -26707.386, -~
$ PROX_CBD.1 <dbl> -120681.94, -149434.22, -259397.77, ~
$ PROX_CHILDCARE.1 <dbl> 319266.925, 441102.177, -120116.816,~
$ PROX_ELDERLYCARE.1 <dbl> -393417.79, 325188.74, 535855.81, 31~
$ PROX_URA_GROWTH_AREA.1 <dbl> -159980.203, -142290.389, -253621.20~
$ PROX_MRT.1 <dbl> -299742.96, -2510522.23, -936853.28,~
$ PROX_PARK.1 <dbl> -172104.47, 523379.72, 209099.85, -7~
$ PROX_PRIMARY_SCH.1 <dbl> 242668.03, 1106830.66, 571462.33, 31~
$ PROX_SHOPPING_MALL.1 <dbl> 300881.390, -87693.378, -126732.712,~
$ PROX_BUS_STOP.1 <dbl> 1210615.44, 1843587.22, 1411924.90, ~
$ NO_Of_UNITS.1 <dbl> 104.8290640, -288.3441183, -9.553294~
$ FAMILY_FRIENDLY.1 <dbl> -9075.370, 310074.664, 5949.746, 155~
$ FREEHOLD.1 <dbl> 303955.61, 396221.27, 168821.75, 121~
$ y <dbl> 3000000, 3880000, 3325000, 4250000, ~
$ yhat <dbl> 2886531.8, 3466801.5, 3616527.2, 543~
$ residual <dbl> 113468.16, 413198.52, -291527.20, -1~
$ CV_Score <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ~
$ Stud_residual <dbl> 0.38207013, 1.01433140, -0.83780678,~
$ Intercept_SE <dbl> 516105.5, 488083.5, 963711.4, 444185~
$ AREA_SQM_SE <dbl> 823.2860, 825.2380, 988.2240, 617.40~
$ AGE_SE <dbl> 5889.782, 6226.916, 6510.236, 6010.5~
$ PROX_CBD_SE <dbl> 37411.22, 23615.06, 56103.77, 469337~
$ PROX_CHILDCARE_SE <dbl> 319111.1, 299705.3, 349128.5, 304965~
$ PROX_ELDERLYCARE_SE <dbl> 120633.34, 84546.69, 129687.07, 1271~
$ PROX_URA_GROWTH_AREA_SE <dbl> 56207.39, 76956.50, 95774.60, 470762~
$ PROX_MRT_SE <dbl> 185181.3, 281133.9, 275483.7, 279877~
$ PROX_PARK_SE <dbl> 205499.6, 229358.7, 314124.3, 227249~
$ PROX_PRIMARY_SCH_SE <dbl> 152400.7, 165150.7, 196662.6, 240878~
$ PROX_SHOPPING_MALL_SE <dbl> 109268.8, 98906.8, 119913.3, 177104.~
$ PROX_BUS_STOP_SE <dbl> 600668.6, 410222.1, 464156.7, 562810~
$ NO_Of_UNITS_SE <dbl> 218.1258, 208.9410, 210.9828, 361.77~
$ FAMILY_FRIENDLY_SE <dbl> 131474.7, 114989.1, 146607.2, 108726~
$ FREEHOLD_SE <dbl> 115954.0, 130110.0, 141031.5, 138239~
$ Intercept_TV <dbl> 3.9720784, 3.3460017, 3.5629010, 0.5~
$ AREA_SQM_TV <dbl> 11.614302, 20.087361, 13.247868, 33.~
$ AGE_TV <dbl> -1.6154474, -9.3441881, -4.1023685, ~
$ PROX_CBD_TV <dbl> -3.22582173, -6.32792021, -4.6235352~
$ PROX_CHILDCARE_TV <dbl> 1.000488185, 1.471786337, -0.3440475~
$ PROX_ELDERLYCARE_TV <dbl> -3.2612693, 3.8462625, 4.1319138, 2.~
$ PROX_URA_GROWTH_AREA_TV <dbl> -2.846248368, -1.848971738, -2.64810~
$ PROX_MRT_TV <dbl> -1.61864578, -8.92998600, -3.4007572~
$ PROX_PARK_TV <dbl> -0.83749312, 2.28192684, 0.66565951,~
$ PROX_PRIMARY_SCH_TV <dbl> 1.59230221, 6.70194543, 2.90580089, ~
$ PROX_SHOPPING_MALL_TV <dbl> 2.75358842, -0.88662640, -1.05686949~
$ PROX_BUS_STOP_TV <dbl> 2.0154464, 4.4941192, 3.0419145, 12.~
$ NO_Of_UNITS_TV <dbl> 0.480589953, -1.380026395, -0.045279~
$ FAMILY_FRIENDLY_TV <dbl> -0.06902748, 2.69655779, 0.04058290,~
$ FREEHOLD_TV <dbl> 2.6213469, 3.0452799, 1.1970499, 8.7~
$ Local_R2 <dbl> 0.8846744, 0.8899773, 0.8947007, 0.9~
$ coords.x1 <dbl> 22085.12, 25656.84, 23963.99, 27044.~
$ coords.x2 <dbl> 29951.54, 34546.20, 32890.80, 32319.~
$ geometry <POINT [m]> POINT (22085.12 29951.54), POI~summary(gwr.adaptive$SDF$yhat)
Min. 1st Qu. Median Mean 3rd Qu. Max.
171347 1102001 1385528 1751842 1982307 13887901 Visualising local R2
Interactive display of the point symbols on a map:
tmap_mode("view")
tm_shape(mpsz_svy21)+
tm_polygons(alpha = 0.1) +
tm_shape(condo_resale.sf.adaptive) +
tm_dots(col = "Local_R2",
border.col = "gray60",
border.lwd = 1) +
tm_view(set.zoom.limits = c(11,14))
Setting back tmap mode to “plot”
tmap_mode("plot")
By URA Planning Region
tmap_mode("view")
tm_shape(mpsz_svy21[mpsz_svy21$REGION_N=="CENTRAL REGION", ])+
tm_polygons()+
tm_shape(condo_resale.sf.adaptive) +
tm_bubbles(col = "Local_R2",
size = 0.15,
border.col = "gray60",
border.lwd = 1)
We can see that the points are out of bound. Should ask prof about this.
Setting back tmap mode to “plot”
tmap_mode("plot")